New life-span results for the nonlinear heat equation

Abstract

We obtain new estimates for the existence time of the maximal solutions to the nonlinear heat equation ${\partial }_{t}u-\Delta u=|u{|}^{\alpha }u,\alpha >0$ with initial values in Lebesgue, weighted Lebesgue spaces or measures. Non-regular, sign-changing, as well as non polynomial decaying initial data are considered. The proofs of the lower-bound estimates of life-span are based on the local construction of solutions. The proofs of the upper-bounds exploit a well-known necessary condition for the existence of nonnegative solutions. In addition, we establish new results for life-span using dilation methods and we give new life-span estimates for Hardy-Hénon parabolic equations.